7-1 Using Proportions Recognize and use ratios and proportions.Recognize and use ratios and proportions. Apply the properties of proportions.Apply the properties of proportions.

Ratio A ratio is a comparison of two quantities using division. Example: The number of boys to girls in a class.

Ways to express a ratio a/b a to b a:b

Example 1 A baseball player’s batting average is the ratio of the number of base hits to the number of at-bats, not including walks. Minnesota Twins’ Joe Mauer had the highest batting average in Major League Baseball in If he had 521 official at-bats and 181 hits, find his batting average.

Number of hits = 181 = Number of at-bats Joe Mauer’s batting average was 0.347

Extended ratios An extended ratio can be used to compare three or more quantities. The expression a:b:c means that the ratio of the first two quantities is a:b, the ratio of the last two quantities is b:c, and the ratio of the first and last quantities is a:c.

Ex. 2 The ratio of the measures of the angles in a triangle is 3:4:5. Find the measures of the angles. Just as ratio ¾ can be written as 3x/4x or 3x:4x, the extended ratio 3:4:5 can be written as 3x:4x:5x. 3x + 4x + 5x = x = 180 X = 15 So the measures of the angles are 3(15) or 45, 4(15) or 60, and 5(15) or = 180

Proportion An equation stating that two ratios are equal is a proportion. Extreme Means extreme a = c mean mean b d extreme The product of the extremes ad and the product of the means bc are called cross products.

The cross products of a proportion are equal.

Equality of Cross Product For any numbers a and c and any nonzero numbers b and d, a = c b d If and only if ad = bc.

Example 2 Solve 3t – 1 = 7 4 8

Nikki can word process 7 words in 6 seconds. At that rate, how many words can she word process in 3 minutes? Words 7 words = x words Time 6 seconds 180 seconds

Example 3 In a triangle, the ratio of the measures of three sides is 8:7:5. and its perimeter is 240 centimeters. Find the measure of each side of the triangle. 8x + 7x + 5x = x = 240 x = 12

Side 1 = 8x = 8(12) = 96cm Side 2 = 7x = 7(12) = 84cm Side 3 = 5x = 5(12) = 60cm

4. Determine which proportions are equivalent. Explain your reasoning. 7 = x y = 8 y = x 7 = 8 8 y x x y

6. 2 inches on a map represent 150 miles. Find a ratio involving 1 inch. 2 inches = 1 inch 150 miles 75 miles

7. The perimeter of a rectangle is 84 feet. The ratio of the width to the length is 2:5, Find the length and width. P = 2l + 2w 84= 2(5x) + 2(2x) 84 = 10x + 4x 84 = 14x 6 = x Length 5(6) is 30, width 2(6) is 12

Ex. 8 The area of a rectangle is 108cm2. The ratio of the width to the length is 3:4. Find the length and width. A = lw 108 = (3x)(4x) 108 = 12x 2 9 = x 2 3 = x Length 4(3) is 12cm, width 3(3) is 9cm

Solve each proportion by using cross products. x = x = 55 x = 1.57

13 = x 91x = 1274 x = 14

X – 2 = 3 x 8 8x – 16 = 3x - 16 = -5x 3.2 = x

If a 6-foot post casts a shadow that is 8 feet long, how tall is an antenna that casts a 60-foot shadow at the same time?

Class work on page 464, problems 1-16 Homework on page 465, problems even numbers.